Radiative Transfer in Homogeneous Nongray Gases With Nonisotropic Particle Scattering

1974 ◽  
Vol 96 (3) ◽  
pp. 385-390 ◽  
Author(s):  
G. A. Domoto ◽  
W. C. Wang
Keyword(s):  
Radiative Transfer ◽  
Rayleigh Scattering ◽  
Narrow Band ◽  
Perturbation Technique ◽  
Wide Band ◽  
Particle Scattering ◽  
Scattering Coefficients ◽  
Earth’S Atmosphere ◽  
Plane Parallel ◽  
Homogeneous Plane

A perturbation technique is presented to treat the problem of radiative transfer in homogeneous, plane parallel, nongray gases with nonisotropic particle scattering. The technique allows use of nongray narrow-band or wide-band models as well as Mie and Rayleigh scattering coefficients and asymmetry factors. Results are obtained in the form of monochromatic transmittance, reflectance, and absorptance of water clouds typical of those in the earth’s atmosphere.

scholarly journals Microwave Emissivity and Accumulation Rate of Polar Firn

1977 ◽  
Vol 18 (79) ◽  
pp. 195-215 ◽  
Author(s):  
H. Jay Zwally
Keyword(s):  
Radiative Transfer ◽  
Rayleigh Scattering ◽  
Accumulation Rate ◽  
Snow Accumulation ◽  
Empirical Parameter ◽  
Microwave Brightness Temperature ◽  
Scattering Coefficients ◽  
Microwave Emissivity ◽  
Snow Crystal ◽  
The Mean

Abstract Radiative transfer theory is formulated to permit a meaningful definition of emissivity for bulk emitting media such as snow. The emissivity in the Rayleigh-Jeans approximation is then the microwave brightness temperature T B divided by an effective physical temperature 〈T〉. The 〈T〉 is an average of the physical temperature, T(z), weighted by a radiative transfer function ƒ(z). Similarly, where e(z) is the local emittance. An approximate ƒ(z) is used to determine analytically the effects of various absorption coefficients, of scattering coefficients that vary with depth, and of the seasonal variation of T(z). It is shown that a mean emissivity, which is equal to the mean annual T B divided by the mean annual surface temperature T m, is a useful quantity for comparing theory and observations. Snow-crystal size measurements, r(z), at seven locations in Greenland and Antarctica are used to determine the Mie/Rayleigh scattering coefficient γs (z and to calculate the mean emissivities. The observed mean emissivities are determined by a which is the average of 12 monthly Nimbus-5 (1.55 cm) microwave observations, and the Tm measured at the same locations. The calculated emissivities are about one-half of the observed values. The assumption that each snow crystal is an independent and equally effective scatterer, and the use of an approximation to ƒ(z), tend to over-estimate the effect of scattering. Therefore, a parameter multiplying γs (z) is used. The emissivities calculated with a single value of this empirical parameter for all seven locations agree well with the observed emissivities, showing that the microwave emissivity variations of dry polar urn can be characterised as a function of the crystal sizes. One optical depth corresponds to a typical fini depth of 5 m, but significant radiation emanates from up to 30 m. Since r(z) depends on the snow accumulation rate A and T m. the sensitivity of the emissivity to changes in T m or A are estimated using this semi-empirical theory. The results show that a one degree change or uncertainty in Tm is approximately equivalent to a 10% change in A, and that such a change will affect the emissivity by 0.003 to 0.014 or the T B by about 0.6 K to 3 K, depending on the location.

scholarly journals Microwave Emissivity and Accumulation Rate of Polar Firn

1977 ◽  
Vol 18 (79) ◽  
pp. 195-215 ◽  
Author(s):  
H. Jay Zwally
Keyword(s):  
Radiative Transfer ◽  
Rayleigh Scattering ◽  
Accumulation Rate ◽  
Snow Accumulation ◽  
Empirical Parameter ◽  
Microwave Brightness Temperature ◽  
Scattering Coefficients ◽  
Microwave Emissivity ◽  
Snow Crystal ◽  
The Mean

AbstractRadiative transfer theory is formulated to permit a meaningful definition of emissivity for bulk emitting media such as snow. The emissivity in the Rayleigh-Jeans approximation is then the microwave brightness temperature TB divided by an effective physical temperature 〈T〉. The 〈T〉 is an average of the physical temperature, T(z), weighted by a radiative transfer function ƒ(z). Similarly, where e(z) is the local emittance. An approximate ƒ(z) is used to determine analytically the effects of various absorption coefficients, of scattering coefficients that vary with depth, and of the seasonal variation of T(z). It is shown that a mean emissivity, which is equal to the mean annual TB divided by the mean annual surface temperature Tm, is a useful quantity for comparing theory and observations. Snow-crystal size measurements, r(z), at seven locations in Greenland and Antarctica are used to determine the Mie/Rayleigh scattering coefficient γs(z) and to calculate the mean emissivities. The observed mean emissivities are determined by a which is the average of 12 monthly Nimbus-5 (1.55 cm) microwave observations, and the Tm measured at the same locations. The calculated emissivities are about one-half of the observed values. The assumption that each snow crystal is an independent and equally effective scatterer, and the use of an approximation to ƒ(z), tend to over-estimate the effect of scattering. Therefore, a parameter multiplying γs(z) is used. The emissivities calculated with a single value of this empirical parameter for all seven locations agree well with the observed emissivities, showing that the microwave emissivity variations of dry polar urn can be characterised as a function of the crystal sizes. One optical depth corresponds to a typical fini depth of 5 m, but significant radiation emanates from up to 30 m. Since r(z) depends on the snow accumulation rate A and Tm. the sensitivity of the emissivity to changes in Tm or A are estimated using this semi-empirical theory. The results show that a one degree change or uncertainty in Tm is approximately equivalent to a 10% change in A, and that such a change will affect the emissivity by 0.003 to 0.014 or the TB by about 0.6 K to 3 K, depending on the location.

Prediction of Radiative Transport in Nonhomogeneous Combustion Gas Mixtures With a Wide Band G(K) Model

2013 ◽  
Author(s):  
Liping Liu ◽  
Jing He
Keyword(s):  
Radiative Transfer ◽  
Narrow Band ◽  
Gaussian Quadrature ◽  
Wide Band ◽  
Gas Mixtures ◽  
Source Distribution ◽  
Distribution Model ◽  
Band Model ◽  
Radiative Transport ◽  
Combustion Gas

A wide band cumulative absorption coefficient distribution, g(k), model is adopted to predict radiative transport in combustion gas mixtures. Prior research has demonstrated similar accuracy of the model to the statistical narrow-band model and superiority to the exponential wideband model under isothermal and homogeneous conditions. This study aims to assess its usefulness in nonhomogeneous media. Sample calculations are performed in a 1D planar slab containing H2O/CO2 mixtures. The six-flux discrete ordinate method (S6-DOM) is employed to solve the radiative transfer equation (RTE), followed by an eight-point Gaussian quadrature of moments with zeroth-order fit. Predictions on the radiative source distribution along the slab and the net radiative flux at the walls are compared to the benchmark line-by-line calculation (LBL) and the statistical narrow-band correlated-k distribution model using the 7-point Gauss-Lobatto quadrature scheme (SNBCK-7). The differences between the g(k) model and LBL are below 5% for a large domain of the layer, with a CPU reduction by a factor of over 30 compared to SNBCK-7 and on the order of 104∼105 compared to LBL. The wide band g(k) model shows significant promise as an accurate and efficient tool to predict radiative transfer in nonhomogenerous media for combustion and fire simulations.

scholarly journals Radiative Transfer Modeling of Microwave Emission and Dependence on Firn Properties

1982 ◽  
Vol 3 ◽  
pp. 54-58 ◽  
Author(s):  
J. C. Comiso ◽  
H. J. Zwally ◽  
J. L. Saba
Keyword(s):  
Radiative Transfer ◽  
Rayleigh Scattering ◽  
Index Of Refraction ◽  
Microwave Emission ◽  
Scattering Coefficients ◽  
Brightness Temperatures ◽  
Significant Temperature ◽  
Radiative Transfer Modeling ◽  
Seasonal Dependence ◽  
Good Agreement

The microwave emission from a model polar firn was calculated using a numerical solution of the radiative transfer equation that included angledependent Rayleigh scattering. The depth-dependent parameters in the equation were physical temperature and the coefficients of scattering and absorption. The coefficients were based on Rayleigh scattering from the snow grains. The bulk emissivity and the seasonal dependence of brightness temperature were calculated for seven locations at which grain sizes were measured as a function of depth. When the absorption and scattering coefficients are adjusted, the modeled emissivities agree with observed emissivities at these locations. The modeled seasonal dependence of brightness temperatures also compares well with values obtained at 1.55 cm wavelength by the Nimbus-5 satellite. Good agreement with data did not occur when the imaginary part of the index of refraction (and, hence, the absorption coefficient) had a significant temperature dependence between 210 and 250 K.

The transport equation of radiative transfer with isotropic scattering

1969 ◽  
Vol 65 (1) ◽  
pp. 199-208 ◽  
Author(s):  
G. E. Hunt
Keyword(s):  
Integral Equation ◽  
Radiative Transfer ◽  
Singular Integral Equation ◽  
Transport Equation ◽  
Singular Integral ◽  
Source Function ◽  
Three Dimensional ◽  
Isotropic Scattering ◽  
Plane Parallel ◽  
Homogeneous Plane

AbstractThe kernel of the integral equation for the source function in a three-dimensional homogeneous atmosphere possesses the properties of a Green's function. These properties are used to transform the integral equation into a singular integral equation for the kernel. The particular case of a homogeneous plane parallel atmosphere is discussed and a solution to the kernel equation is obtained at all points of the atmosphere.

scholarly journals Radiative Transfer Modeling of Microwave Emission and Dependence on Firn Properties

1982 ◽  
Vol 3 ◽  
pp. 54-58 ◽  
Author(s):  
J. C. Comiso ◽  
H. J. Zwally ◽  
J. L. Saba
Keyword(s):  
Radiative Transfer ◽  
Rayleigh Scattering ◽  
Index Of Refraction ◽  
Microwave Emission ◽  
Scattering Coefficients ◽  
Brightness Temperatures ◽  
Significant Temperature ◽  
Radiative Transfer Modeling ◽  
Seasonal Dependence ◽  
Good Agreement

The microwave emission from a model polar firn was calculated using a numerical solution of the radiative transfer equation that included angledependent Rayleigh scattering. The depth-dependent parameters in the equation were physical temperature and the coefficients of scattering and absorption. The coefficients were based on Rayleigh scattering from the snow grains. The bulk emissivity and the seasonal dependence of brightness temperature were calculated for seven locations at which grain sizes were measured as a function of depth. When the absorption and scattering coefficients are adjusted, the modeled emissivities agree with observed emissivities at these locations. The modeled seasonal dependence of brightness temperatures also compares well with values obtained at 1.55 cm wavelength by the Nimbus-5 satellite. Good agreement with data did not occur when the imaginary part of the index of refraction (and, hence, the absorption coefficient) had a significant temperature dependence between 210 and 250 K.

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